In GR, where \(d_L^{\,\rm gw}(z)=d_L^{\,\rm em}(z)\equiv d_L(z)\) , the GW observation gives \(d_L(z)\)  and, for a source with electromagnetic counterpart, we also have the redshift \(z\)  of the source. We can then use this to extract \(H_0\) , as already successfully done in for
GW170817 [1]}. The error from this single detection is still too large to discriminate between the value of \(H_0\)  obtained from early and late-Universe probes, and one can estimate that \({\cal O}(50-100)\)  standard sirens with counterpart are needed to reach the accuracy required to arbitrate this discrepancy [2]}, [3]}.
Binary black holes (BBHs), in general, are not expected to have an electromagnetic counterpart, and also for BNS only a small fraction of relatively close sources is expected to have an observed counterpart.
However, several statistical methods have been investigated to extract cosmological information from an ensemble of detections. In this context, compact binary coalescences without electromagnetic counterpart are often called `dark sirens'. The first and most widely studied possibility is to correlate
the GW signals with galaxy catalogs [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, and has been applied to extract \(H_0\)  from the recent LIGO/Virgo detections in a number of papers [14]}, [15]}, [16]}, [17]}, [18]}, [19]}. Another option, proposed more recently, is to use multiply lensed GW events to identify, or restrict, the host galaxy [20]}, [21]}.
It is, however, of great interest to explore possibilities that do not make use of any electromagnetic information, not even in the form of a galaxy catalog. This is particularly relevant for events at high redshift, where the completeness of catalogs and the possibility of detecting a counterpart can be a significant limitation. Since, as we will see, the effect of \(\Xi _0\)  cumulates with the propagation, this is particularly important for testing modified gravity.
